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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000486
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2,3,1,6,5] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St001095
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001095: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001095: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Values
[1,1] => 11 => 11 => ([(0,2),(2,1)],3)
=> 0
[2] => 10 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,1,1] => 111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2] => 110 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,1] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0
[3] => 100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0
[1,1,1,1] => 1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,2] => 1110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[1,2,1] => 1101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
[1,3] => 1100 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0
[2,1,1] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[2,2] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0
[3,1] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[4] => 1000 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[1,1,1,1,1] => 11111 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,2] => 11110 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1
[1,1,2,1] => 11101 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1
[1,1,3] => 11100 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[1,2,1,1] => 11011 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[1,2,2] => 11010 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[1,3,1] => 11001 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1
[1,4] => 11000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1
[2,1,1,1] => 10111 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[2,1,2] => 10110 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[2,2,1] => 10101 => 01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 0
[2,3] => 10100 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[3,1,1] => 10011 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1
[3,2] => 10010 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[4,1] => 10001 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[5] => 10000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[1,1,1,1,1,1] => 111111 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0
[1,1,1,1,2] => 111110 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1
[1,1,1,2,1] => 111101 => 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 1
[1,1,1,3] => 111100 => 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0
[1,1,2,1,1] => 111011 => 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[1,1,2,2] => 111010 => 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 1
[1,1,3,1] => 111001 => 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 1
[1,1,4] => 111000 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1
[1,2,1,1,1] => 110111 => 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[1,2,1,2] => 110110 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 2
[1,2,2,1] => 110101 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 1
[1,2,3] => 110100 => 011010 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 1
[1,3,1,1] => 110011 => 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 1
[1,3,2] => 110010 => 010110 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 0
[1,4,1] => 110001 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1
[1,5] => 110000 => 001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 0
[2,1,1,1,1] => 101111 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[2,1,1,2] => 101110 => 011110 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1
[2,1,2,1] => 101101 => 011101 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 1
[2,1,3] => 101100 => 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 1
[2,2,1,1] => 101011 => 011011 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 0
[2,2,2] => 101010 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 0
[2,3,1] => 101001 => 100101 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 1
[2,4] => 101000 => 010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 0
[3,1,1,1] => 100111 => 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 1
[3,1,2] => 100110 => 001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1
[3,2,1] => 100101 => 001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 1
[3,3] => 100100 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 0
Description
The number of non-isomorphic posets with precisely one further covering relation.
Matching statistic: St001942
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001942: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001942: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 67%
Values
[1,1] => 11 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2] => 10 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1] => 111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2] => 110 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1] => 101 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[3] => 100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1] => 1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,2] => 1110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,1] => 1101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[1,3] => 1100 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 0 + 1
[2,1,1] => 1011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[2,2] => 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
[3,1] => 1001 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[4] => 1000 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[1,1,1,1,1] => 11111 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,2] => 11110 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[1,1,2,1] => 11101 => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[1,1,3] => 11100 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0 + 1
[1,2,1,1] => 11011 => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[1,2,2] => 11010 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
[1,3,1] => 11001 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1 + 1
[1,4] => 11000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 1 + 1
[2,1,1,1] => 10111 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[2,1,2] => 10110 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
[2,2,1] => 10101 => 01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 0 + 1
[2,3] => 10100 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
[3,1,1] => 10011 => 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
[3,2] => 10010 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[4,1] => 10001 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0 + 1
[5] => 10000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[1,1,1,1,1,1] => 111111 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 0 + 1
[1,1,1,1,2] => 111110 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1 + 1
[1,1,1,2,1] => 111101 => 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 1 + 1
[1,1,1,3] => 111100 => 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0 + 1
[1,1,2,1,1] => 111011 => 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1 + 1
[1,1,2,2] => 111010 => 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 1 + 1
[1,1,3,1] => 111001 => 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 1 + 1
[1,1,4] => 111000 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1 + 1
[1,2,1,1,1] => 110111 => 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1 + 1
[1,2,1,2] => 110110 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 2 + 1
[1,2,2,1] => 110101 => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 1 + 1
[1,2,3] => 110100 => 011010 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 1 + 1
[1,3,1,1] => 110011 => 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 1 + 1
[1,3,2] => 110010 => 010110 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 0 + 1
[1,4,1] => 110001 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? = 1 + 1
[1,5] => 110000 => 001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 0 + 1
[2,1,1,1,1] => 101111 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[2,1,1,2] => 101110 => 011110 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1 + 1
[2,1,2,1] => 101101 => 011101 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 1 + 1
[2,1,3] => 101100 => 110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 1 + 1
[2,2,1,1] => 101011 => 011011 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 0 + 1
[2,2,2] => 101010 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? = 0 + 1
[2,3,1] => 101001 => 100101 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 1 + 1
[2,4] => 101000 => 010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 0 + 1
[3,1,1,1] => 100111 => 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 1 + 1
[3,1,2] => 100110 => 001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1 + 1
[3,2,1] => 100101 => 001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? = 1 + 1
[3,3] => 100100 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 0 + 1
Description
The number of loops of the quiver corresponding to the reduced incidence algebra of a poset.
Matching statistic: St000091
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,1] => [1,0,1,0]
=> 1010 => [1,1,1,1] => 0
[2] => [1,1,0,0]
=> 1100 => [2,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => 0
[1,2] => [1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => 1
[2,1] => [1,1,0,0,1,0]
=> 110010 => [2,2,1,1] => 0
[3] => [1,1,1,0,0,0]
=> 111000 => [3,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => ? = 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => ? = 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => [1,1,2,2,1,1] => ? = 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => ? = 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => [2,2,1,1,1,1] => ? = 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => [2,2,2,2] => ? = 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => [3,3,1,1] => ? = 1
[4] => [1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => ? = 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,1,1,1,1,1,1,1,1,1] => ? = 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,1,1,1,1,1,2,2] => ? = 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => [1,1,1,1,2,2,1,1] => ? = 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,1,1,1,3,3] => ? = 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => [1,1,2,2,1,1,1,1] => ? = 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => [1,1,2,2,2,2] => ? = 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => [1,1,3,3,1,1] => ? = 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => [1,1,4,4] => ? = 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => [2,2,1,1,1,1,1,1] => ? = 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => [2,2,1,1,2,2] => ? = 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => [2,2,2,2,1,1] => ? = 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => [2,2,3,3] => ? = 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => [3,3,1,1,1,1] => ? = 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => [3,3,2,2] => ? = 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => [4,4,1,1] => ? = 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => [5,5] => ? = 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => ? = 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => [1,1,1,1,1,1,1,1,2,2] => ? = 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 101010110010 => [1,1,1,1,1,1,2,2,1,1] => ? = 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => [1,1,1,1,1,1,3,3] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 101011001010 => [1,1,1,1,2,2,1,1,1,1] => ? = 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 101011001100 => [1,1,1,1,2,2,2,2] => ? = 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 101011100010 => [1,1,1,1,3,3,1,1] => ? = 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => [1,1,1,1,4,4] => ? = 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 101100101010 => [1,1,2,2,1,1,1,1,1,1] => ? = 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 101100101100 => [1,1,2,2,1,1,2,2] => ? = 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 101100110010 => [1,1,2,2,2,2,1,1] => ? = 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 101100111000 => [1,1,2,2,3,3] => ? = 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 101110001010 => [1,1,3,3,1,1,1,1] => ? = 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 101110001100 => [1,1,3,3,2,2] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => [1,1,4,4,1,1] => ? = 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => [1,1,5,5] => ? = 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => [2,2,1,1,1,1,1,1,1,1] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 110010101100 => [2,2,1,1,1,1,2,2] => ? = 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => [2,2,1,1,2,2,1,1] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => [2,2,1,1,3,3] => ? = 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 110011001010 => [2,2,2,2,1,1,1,1] => ? = 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => [2,2,2,2,2,2] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 110011100010 => [2,2,3,3,1,1] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => [2,2,4,4] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => [3,3,1,1,1,1,1,1] => ? = 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => [3,3,1,1,2,2] => ? = 1
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St000709
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000709: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000709: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 0
[2] => [1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0
[1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 1
[2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 0
[3] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? = 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? = 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? = 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? = 1
[4] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ? = 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => ? = 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => ? = 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? = 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => ? = 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => ? = 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? = 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? = 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => ? = 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => ? = 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ? = 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? = 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? = 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? = 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? = 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => ? = 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => ? = 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => ? = 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => ? = 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => ? = 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => ? = 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => ? = 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => ? = 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => ? = 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> [2,1,6,7,8,5,4,3,10,9,12,11] => ? = 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> [2,1,6,7,8,5,4,3,11,12,10,9] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> [2,1,7,8,9,10,6,5,4,3,12,11] => ? = 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? = 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => ? = 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => ? = 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => ? = 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> [4,5,6,3,2,1,8,7,10,9,12,11] => ? = 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> [4,5,6,3,2,1,8,7,11,12,10,9] => ? = 1
Description
The number of occurrences of 14-2-3 or 14-3-2.
The number of permutations avoiding both of these patterns is the case $k=2$ of the third item in Corollary 34 of [1].
Matching statistic: St001868
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,1] => [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0
[2] => [1,1,0,0]
=> [2,3,1] => [2,3,1] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 1
[2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => [2,4,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ? = 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? = 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => ? = 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [2,3,5,1,4] => ? = 1
[4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => ? = 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,6,3,5] => ? = 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? = 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,2,4,5] => ? = 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,2,6,4] => ? = 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,4,6,2,5] => ? = 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? = 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,3,4,5] => ? = 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,5,1,3,6,4] => ? = 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,4,1,6,3,5] => ? = 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,4,1,5,6,3] => ? = 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? = 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [2,3,5,1,6,4] => ? = 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [2,3,4,6,1,5] => ? = 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,2,3,4,7,5] => ? = 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => ? = 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,2,3,6,7,4] => ? = 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,2,7,3,5,6] => ? = 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,2,6,3,7,5] => ? = 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [4,1,2,5,7,3,6] => ? = 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,2,5,6,7,3] => ? = 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [3,1,7,2,4,5,6] => ? = 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,6,2,4,7,5] => ? = 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,1,5,2,7,4,6] => ? = 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,5,2,6,7,4] => ? = 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,1,4,7,2,5,6] => ? = 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [3,1,4,6,2,7,5] => ? = 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,4,5,7,2,6] => ? = 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? = 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,6,1,3,4,7,5] => ? = 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => [2,5,1,3,7,4,6] => ? = 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,5,1,3,6,7,4] => ? = 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => [2,4,1,7,3,5,6] => ? = 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => [2,4,1,6,3,7,5] => ? = 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => [2,4,1,5,7,3,6] => ? = 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,4,1,5,6,7,3] => ? = 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => [2,3,7,1,4,5,6] => ? = 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,3,6,1,4,7,5] => ? = 1
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Matching statistic: St000260
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1,1] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> ? = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ? = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000456
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1,1] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> ? = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? = 0 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ? = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001722
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => ? = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => ? = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => ? = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => ? = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => ? = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => ? = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => ? = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => ? = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 110110001100 => ? = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 110011011000 => ? = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 110111001000 => ? = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 110011001010 => ? = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 110110001010 => ? = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 110011010100 => ? = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 110111010000 => ? = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 110011100100 => ? = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 110110100100 => ? = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => ? = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 110111100000 => ? = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 110011101000 => ? = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 110110101000 => ? = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 111001001100 => ? = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => ? = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 111001011000 => ? = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 111101001000 => ? = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 111001001010 => ? = 0 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? = 0 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => ? = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 111101010000 => ? = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => ? = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 111110000100 => ? = 1 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
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